Robust Control Design Using H-∞ Methods

In designing a robust control system, one must specify the class of uncertainties the control system is to be robust against. Within the modern control framework, one approach to designing robust control systems is to begin with a plant model which not only models the nominal plant behavior but also models the type of uncertainties which are expected. Such a plant model is referred to as an uncertain system.

As mentioned in Chapter 1, the uncertain system models we will deal with in this book originate from the linear fractional transformation shown in Figure 1.1.1 on page 2 and Figure 1.2.2 on page 10. In Chapter 1, we discussed two typical bounds on the uncertainty Δ in the uncertain systems shown in these figures. This allows one to describe uncertainties arising either in the form of time-varying normbounded blocks satisfying (1.1.1) or in the form of transfer function blocks in which the H^∞norm is bounded as in condition (1.1.2). The H^∞-norm bound uncertainty description is closely related to the frequency domain integral quadratic constraint (IQC) uncertainty description. The limitations of these two forms of uncertainty description were discussed briefly in Chapter 1.

This chapter contains a number of important results related to H^∞ control theory, risk sensitive control theory, algebraic Riccati equations and uncertain systems with norm-bounded uncertainty. These results will be useful when addressing the problems of optimal guaranteed cost control and minimax optimal control.

In this chapter, we present a collection of results concerning the so called S-procedure. The term “S-procedure” was introduced by Aizerman and Gantmacher in the monograph [2] (see also [65]) to denote a method which had been frequently used in the area of nonlinear control; e.g., see [119, 112, 169]. More recently, similar results have been extensively used in the Western control theory literature and the name S-procedure has remained; e.g., see [127, 185, 190, 210, 26]. As mentioned in Section 1.2, a feature of the S-procedure approach is that it allows for non-conservative results to be obtained for control problems involving structured uncertainty. In fact, the S-procedure provides a method for converting robust control problems involving structured uncertainty into parameter dependent problems involving unstructured uncertainty. Furthermore, S-procedure methods find application to many robust control problems not included in this book; e.g., see [196, 131, 202]. A general and systematic description of the S-procedure can be found, for example, in the monograph [67].

As mentioned in Chapter 1, one approach to the robust linear quadratic regulator problem takes as its launching point the Riccati equation approach to the quadratic stabilization of uncertain systems and its connection to H^∞ control; see [150, 143, 102, 252, 253, 41]. In these papers, the controller is obtained by solving a certain game type Riccati equation arising in an associated H^∞ control problem. The solution to the Riccati equation also defines a fixed quadratic Lyapunov function for the closed loop uncertain system. Hence, the controller obtained using this approach leads to a quadratically stable closed loop system. This approach can also be used to obtain a controller which not only guarantees robust stability but also guarantees a certain level of robust performance. In this case, the quadratic Lyapunov function is used to give an upper bound on the closed loop value of a quadratic cost functional; e.g., see [35, 13, 14]. Such a controller is referred to as a quadratic guaranteed cost controller. We consider the problem of quadratic guaranteed cost control in Section 5.2. The uncertainties under consideration in Section 5.2 are norm-bounded uncertainties. The main result to be presented on this problem is a Riccati equation approach to the construction of an optimal state-feedback quadratic guaranteed cost control. This result was originally obtained by Petersen and McFarlane in references [155, 154, 156]. Note that the result of Petersen and McFarlane goes beyond the earlier results of [35, 13, 14] in that it leads to an “optimal” quadratic guaranteed cost controller whereas the earlier results were sub-optimal in this sense. That is, for the case of norm-bounded uncertainty, the Riccati equation approach presented in Section 5.2 yields an upper bound on the closed loop value of the cost functional which is at least as good as any that can be obtained via the use of a fixed quadratic Lyapunov function.

As mentioned in Chapter 1, the problem of directly extending the results of Chapter 5 to the case of a finite time horizon or to the case of time-varying uncertain systems, appears to be mathematically intractable. In this chapter, we overcome this difficulty by considering an alternative class of uncertain systems with structured uncertainty.

In this chapter, we revisit the classical Lur’e-Postnikov approach to the stability of nonlinear control systems. The problems considered in this chapter include the problem of absolute stability and also its extensions such as the problem of absolute stabilization and the problem of structured dissipativity. As in previous chapters of this book, the approach taken here is to convert the problem under consideration into an equivalent H^∞ control problem. Such a conversion is accomplished by making use of the corresponding S-procedure theorem.

The results presented in this chapter extend the notions of minimax optimal control and absolute stabilization to the realm of stochastic uncertain systems. Some motivation for this extension was given in the Chapter 1 and in Section 2.4. In particular, in Subsections 2.4.1 and 2.4.2, some examples where given of uncertain systems which led naturally to descriptions in terms of stochastic processes. Also, Section 2.4 introduced definitions of stochastic uncertain system which provided a stochastic uncertain system framework for stochastic systems with multiplicative noise and additive noise. In this chapter, we address minimax optimal control problems for the stochastic uncertain systems introduced in Section 2.4.

This chapter considers stabilization problems for uncertain linear systems containing unstructured or structured uncertainty described by integral quadratic constraints. In these problems, we will be interested in the question of whether any advantage can be obtained via the use of a nonlinear controller as opposed to a linear controller. This question is of interest since if no advantage can be obtained via the use of nonlinear controller, then there can be little motivation for the use of a complicated nonlinear controller such as adaptive controller.

As mentioned in Chapter 1, one of the most attractive methods for designing multi-input multi-output feedback control systems is the linear quadratic Gaussian (LQG) method; e.g., see [3, 46]. In particular, the use of a quadratic cost function is well motivated in many control problems. Also, the stochastic white noise model is often a good approximation to the noise found in practical control problems. These facts, together with the fact that the LQG control problem can be solved via reliable numerical techniques based on the algebraic Riccati equation, provide good motivation for the use of the LQG method. However, the LQG design technique does not address the issue of robustness and it is known that a controller designed using the LQG technique can have arbitrarily poor robustness properties; e.g., see [50]. Since, the issue of robustness is critical in most control system design problems, we have been motivated to consider the problem of obtaining a robust version of the LQG control technique. These results were presented in Section 8.5; see also [206,49, 134, 72, 146, 152, 230, 231]. In particular, the results of Section 8.5 generalize the LQG problem to a robust LQG problem for stochastic uncertain systems. In this chapter, we will apply the results of Section 8.5 to the problem of designing a missile autopilot.

This chapter considers the problem of active noise control in an acoustic duct. The problem of active noise control in a duct has been the subject of a large amount of research; e.g., see [63, 86]. We consider a broadband feedback control approach to active noise control rather than the narrowband adaptive feedforward approach taken in many early papers on active noise control. The advantage of the feedback control approach is that it can handle practical noise reduction problems in which the noise source has a broad spectrum. The disadvantage of the feedback control approach is that it opens up the possibility of instability and the control system must be carefully designed to give adequate robustness. Also, it usually necessitates the construction of a model for the duct.